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- W4299953092 abstract "Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any $lambda in (0,1)$ there exists a constant $C_lambda$ such that every closed Riemannian surface of genus $g$ whose area is normalized at $4pi(g-1)$ has at least $[lambda g]$ homologically independent loops of length at most $C_lambda log(g)$. This result extends Gromov's asymptotic $log(g)$ bound on the homological systole of genus $g$ surfaces. We construct hyperbolic surfaces showing that our general result is sharp. We also extend the upper bound obtained by P. Buser and P. Sarnak on the minimal norm of nonzero period lattice vectors of Riemann surfaces %systole of Jacobians of Riemann surfaces in their geometric approach of the Schottky problem to almost $g$ homologically independent vectors. Then, we consider the lengths of pants decompositions on complete Riemannian surfaces in connexion with Bers' constant and its generalizations. In particular, we show that a complete noncompact Riemannian surface of genus $g$ with $n$ ends and area normalized to $4pi (g+frac{n}{2}-1)$ admits a pants decomposition whose total length (sum of the lengths) does not exceed $C_g , n log (n+1)$ for some constant $C_g$ depending only on the genus. Finally, we obtain a lower bound on the systolic area of finitely presentable nontrivial groups with no free factor isomorphic to $Z$ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal." @default.
- W4299953092 created "2022-10-03" @default.
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- W4299953092 date "2010-11-12" @default.
- W4299953092 modified "2023-09-28" @default.
- W4299953092 title "Short loop decompositions of surfaces and the geometry of Jacobians" @default.
- W4299953092 doi "https://doi.org/10.48550/arxiv.1011.2962" @default.
- W4299953092 hasPublicationYear "2010" @default.
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